Abstract
We study the extension problem of a given sequence defined by a finite order recurrence to a sequence defined by an infinite order recurrence with periodic coefficient sequence. We also study infinite order recurrence relations in a strong sense and give a complete answer to the extension problem. We also obtain a Binet-type formula, answering several open questions about these sequences and their characteristic power series.
Highlights
The notion of an ∞-generalized Fibonacci sequence (∞-GFS) has been introduced in [7] and studied in [1, 8, 10]. This class of sequences defined by linear recurrences of infinite order is an extension of the class of ordinary r-generalized Fibonacci sequences (r-GFSs) with r finite defined by linear recurrences of rth order
Given an r-GFS, can one always extend it to an ∞-GFS associated with a periodic coefficient sequence? If it is not always the case, characterize those r-GFSs which can be extended to an ∞-GFS associated with a periodic coefficient sequence
The main result of this paper is a characterization theorem of those r-GFSs which can be extended to a strongly ∞GFS associated with a periodic coefficient sequence (Theorem 3.2)
Summary
The notion of an ∞-generalized Fibonacci sequence (∞-GFS) has been introduced in [7] and studied in [1, 8, 10]. If it is not always the case, characterize those r-GFSs which can be extended to an ∞-GFS associated with a periodic coefficient sequence. If T(x) does not have any root ξ ∈ C with ξr = 1, there exists a sequence {V−n}∞n=0 such that {Vn}n∈Z is an ∞-GFS associated with the periodic coefficient sequence (2.2).
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